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arXiv · 2603.00571

The Bombieri--van der Poorten Formula for Partial Quotients of Higher Degree Algebraic Irrationals

Karsten Müller
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Abstract

The fundamental relationship between the partial quotients $b_{n+1}$ of an algebraic irrational $\alpha = \sqrt[m]{k}$ and its corresponding algebraic form $d_n = |p_n^m - k q_n^m|$ was elegantly proposed by Bombieri and van der Poorten. In this paper, we work out the explicit analytical details of the framework for any degree $m \geq 3$. We provide a closed-form derivation of the error term and prove for the cubic case that the remainder $|R_n|$ is strictly bounded by 1 for all convergents with $q_n \geq 2$.

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Karsten Müller. 2026-02-28. The Bombieri--van der Poorten Formula for Partial Quotients of Higher Degree Algebraic Irrationals. https://arxiv.org/abs/2603.00571

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